Manas Rachh

Fast algorithms for Quadrature by Expansion I: Globally valid expansions

Abstract The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast algorithms for solving the resulting dense linear systems. Classically, these tools were developed separately. In this work, we present a unified numerical scheme based on coupling Quadrature by Expansion , a recent quadrature method, to a customized Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. The method allows the evaluation of layer potentials in linear-time complexity, anywhere in space, with a uniform, user-chosen level of accuracy as a black-box computational method. Providing this capability requires geometric and algorithmic considerations beyond the needs of standard FMMs as well as careful consideration of the accuracy of multipole translations. We illustrate the speed and accuracy of our method with various numerical examples.

Integral Equation Methods for Elastance and Mobility Problems in Two Dimensions

We present new integral representations in two dimensions for the elastance problem in electrostatics and the mobility problem in Stokes flow. These representations lead to resonance-free Fredholm integral equations of the second kind and well conditioned linear systems upon discretization. By coupling our integral equations with high order quadrature and fast multipole acceleration, large-scale problems can be solved with only modest computing resources. We also discuss some applications of these boundary value problems in applied physics.

An integral equation formulation for rigid bodies in Stokes flow in three dimensions

Abstract We present a new derivation of a boundary integral equation (BIE) for simulating the three-dimensional dynamics of arbitrarily-shaped rigid particles of genus zero immersed in a Stokes fluid, on which are prescribed forces and torques. Our method is based on a single-layer representation and leads to a simple second-kind integral equation. It avoids the use of auxiliary sources within each particle that play a role in some classical formulations. We use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials, so that only a small number of spatial discretization points per particle are required. The resulting discrete sums are computed in O ( n ) time, where n denotes the number of particles, using the fast multipole method (FMM). The particle positions and orientations are updated by a high-order time-stepping scheme. We illustrate the accuracy, conditioning and scaling of our solvers with several numerical examples.

Integral Equation Formulation of the Biharmonic Dirichlet Problem

We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman–Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.

Accurate derivative evaluation for any Grad-Shafranov solver

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantities such as the magnetic field, the parallel current density and the magnetic curvature with much higher accuracy than has been previously feasible on the affordable coarse grids that are usually implemented.

An Efficient High Order Method for Dislocation Climb in Two Dimensions

We present an efficient high order method for dislocation dynamics simulation of vacancy-assisted dislocation climb in two dimensions. The method is based on a second kind integral equation (SKIE) formulation that represents the vacancy concentration via the sum of double layer potentials and point sources located at each dislocation, where the climb velocity of each dislocation (or the strength of each point source) is proportional to the integral of the unknown density on the boundary of each dislocation. The method discretizes the interfaces only. Unlike previously used formulations, the proposed method avoids the need for introducing additional unknowns or integrating kernels with logarithmic singularity, and the boundary integrals in the formulation are easily discretized via the trapezoidal rule with spectral accuracy. Thus, the number of unknowns in the linear system to achieve certain accuracy is optimal for typical settings in dislocation dynamics. We compare three different methods for solving r...

A fluctuating boundary integral method for Brownian suspensions

Abstract We present a fluctuating boundary integral method (FBIM) for overdamped Brownian Dynamics (BD) of two-dimensional periodic suspensions of rigid particles of complex shape immersed in a Stokes fluid. We develop a novel approach for generating Brownian displacements that arise in response to the thermal fluctuations in the fluid. Our approach relies on a first-kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface, with zero mean and covariance proportional to the Greens function for Stokes flow (Stokeslet). This approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics, handles particles of complex shape, achieves high order of accuracy, and can be generalized to three dimensions and other boundary conditions. We show that Brownian displacements generated by our method obey the discrete fluctuation–dissipation balance relation (DFDB). Based on a recently-developed Positively Split Ewald method Fiore et al. (2017) [24] , near-field contributions to the Brownian displacements are efficiently approximated by iterative methods in real space, while far-field contributions are rapidly generated by fast Fourier-space methods based on fluctuating hydrodynamics. FBIM provides the key ingredient for time integration of the overdamped Langevin equations for Brownian suspensions of rigid particles. We demonstrate that FBIM obeys DFDB by performing equilibrium BD simulations of suspensions of starfish-shaped bodies using a random finite difference temporal integrator.

A coarse-grained computational model of the nuclear pore complex predicts Phe-Gly nucleoporin dynamics

The phenylalanine-glycine–repeat nucleoporins (FG-Nups), which occupy the lumen of the nuclear pore complex (NPC), are critical for transport between the nucleus and cytosol. Although NPCs differ in composition across species, they are largely conserved in organization and function. Transport through the pore is on the millisecond timescale. Here, to explore the dynamics of nucleoporins on this timescale, we use coarse-grained computational simulations. These simulations generate predictions that can be experimentally tested to distinguish between proposed mechanisms of transport. Our model reflects the conserved structure of the NPC, in which FG-Nup filaments extend into the lumen and anchor along the interior of the channel. The lengths of the filaments in our model are based on the known characteristics of yeast FG-Nups. The FG-repeat sites also bind to each other, and we vary this association over several orders of magnitude and run 100-ms simulations for each value. The autocorrelation functions of the orientation of the simulated FG-Nups are compared with in vivo anisotropy data. We observe that FG-Nups reptate back and forth through the NPC at timescales commensurate with experimental measurements of the speed of cargo transport through the NPC. Our results are consistent with models of transport where FG-Nup filaments are free to move across the central channel of the NPC, possibly informing how cargo might transverse the NPC.